Prof. Brian L. Evans

Dept. of Electrical and Computer Engineering

The University of Texas at Austin

EE345S Real-Time Digital Signal Processing Lab Fall 2006

Lecture 5 http://courses.utexas.edu/

Finite Impulse Response Filters

5 - 2

Discrete-Time Impulse Signal

• Let [k] be a discrete-time impulse function, a.k.a. Kronecker delta function:

• Impulse response is response of discrete-time LTI system to discrete impulse functionExample: delay by one sample

• Finite impulse response filterNon-zero extent of impulse response is finite

Can be in continuous time or discrete time

Also called tapped delay line (see slides 3-13, 3-19, 5-3)

00

01

k

kk

1z[k] h[k]

]1[][ kkh

5 - 3

1

0

][ ][ ][M

m

mkxmhky

Discrete-time Tapped Delay Line

• Impulse response h[k] of finite extent k = 0,…, M-1

• Block diagram (finite impulse response filter)

z-1 z-1 z-1…

…

x[k]

y[k]

h[0] h[1] h[2] h[M-1]

x[k-1]

Discrete-time convolution

Applications of continuous-time tapped delay lines?

5 - 4

Discrete-time Convolution Derivation

• Output y[k] for input x[k]

• Any signal can be decomposedinto sum of discrete impulses

• Apply linear properties

• Apply shift-invariance

• Apply change of variables

y[k] = h[0] x[k] + h[1] x[k-1]

= ( x[k] + x[k-1] ) / 2k

h[k]

2

1

Averaging filter impulse response

0 1 2 3

kxTky

m

mkmxTky

mkTmxkym

mkhmxkym

mkxmhkym

5 - 5

Comparison to Continuous Time

• Continuous-time convolution of x(t) and h(t)

For each t, compute different (possibly) infinite integral

• In discrete-time, replace integral with summation

For each k, compute different (possibly) infinite summation

• LTI systemFrom impulse response and input, one can determine output

Impulse response uniquely characterizes LTI system

dtxhdthxthtxty

mkxmhmkhmxkymm

5 - 6

Convolution Demos

• Johns Hopkins University Demonstrationshttp://www.jhu.edu/~signals (http://www.jhu.edu/~signals)

Convolution applet to animate convolution of simple signals and hand-sketched signals

Convolving two rectangular pulses of same width gives triangle with width of twice the width of rectangular pulses(also see Appendix E for intermediate calculations)

t

1x(t)

0 Ts 2Ts

Ts

t

1h(t)

0 Ts

* = y(t)

Ts

t

5 - 7

Linear Time-Invariant Systems

• Complex exponentials zk havea special property when theyare input into LTI systems

• Output will be same complexexponential weighted by H(z)

• When we specialize the z-domain to frequency domain, magnitude of H(z) will control which frequencies are attenuated or passed

• H(z) is also known as the transfer function

)(

][

zHz

zmhz

zmh

zkhkfkhky

km

mk

m

mk

k

5 - 8

Linear Time-Invariant Systems

• The Fundamental Theorem of Linear SystemsIf a complex sinusoid were input into an LTI system, then

the output would be a complex sinusoid of the same frequency that has been scaled by the frequency response of the LTI system at that frequency

Scaling may attenuate the signal and shift it in phaseExample in continuous time: see handout FExample in discrete time. Let x[k] = e j k,

H() is discrete-time Fourier transform of h[k]H() is also called the frequency response

Heemhemheky kj

m

mjkj

m

mkj ][][][

H()

5 - 9

Frequency Response

• For continuous-time systems, response to complex sinusoid is

• For discrete-time systems, z-k = (r e j )–k = r-k e – j k and the response is

• For discrete-time systems, response to complex sinusoid is

jHtjHt

ejHe tjtj

cos cos

kk zzHz )(

cos cos

jj

kjjkj

eHkeHk

eeHe

frequency response

frequency response

5 - 10

Example: Ideal Delay

• Continuous TimeDelay by T seconds

Impulse response

Frequency response

• Discrete TimeDelay by 1 sample

Impulse response

Frequency response

)( Ttxty

Tx(t) y(t)

)( Ttth

TjeH

]1[][ kxky

1zx[k] y[k]

]1[][ kkh

jeH

1 || H

TH

1 || H

H

5 - 11

Frequency Response

• System response to complex sinusoid e j t for all possible frequencies where = 2 f :

• Above: passes low frequencies, a.k.a. lowpass filter• FIR filters are only realizable LTI filters that can

have linear phase over all frequencies• Not all FIR filters exhibit linear phase

|H()|

|H()|

p ss p

passband

stopband stopbandLinearphase

THd

ddelay

)()(

5 - 12

Linear Time-Invariant Systems

• Any linear time-invariant system (LTI) system, whether continuous-time or discrete-time, can be uniquely characterized by itsImpulse response: response of system to an impulse ORFrequency response: response of system to a complex

sinusoid (e j t or e j k) for all possible frequencies ORTransfer function: general transform of impulse response

(Laplace transform for continuous-time systems and z-transform for discrete-time systems)

• Given one, we can find other two if they existGive an impulse response that has a Laplace transform but

not a Fourier transform? What about the other way?

5 - 13

Mandrill Demo (DSP First)

• Five-tap discrete-time averaging FIR filter with input x[k] and output y[k]

Lowpass filter (smooth/blur input signal)Impulse response is {1/5, 1/5, 1/5, 1/5, 1/5}

• First-order difference FIR filter

Highpass filter (sharpensinput signal)

Impulse response is {1, -1}

]4[5

1]3[

5

1]2[

5

1]1[

5

1][

5

1 kxkxkxkxkxky

]1[][ kxkxky

k

h[k]

1

First-order difference impulse response

1

5 - 14

Mandrill Demo (DSP First)

• DSP First demos: http://users.ece.gatech.edu/~dspfirst• From lowpass filter to highpass filter

original image blurred image sharpened/blurred image

• From highpass to lowpass filteroriginal image sharpened image blurred/sharpened image

• Frequencies that are zeroed out can never be recovered (e.g. DC is zeroed out by highpass filter)

• Order of two LTI systems in cascade can be switched under the assumption that computations are performed in exact precision

5 - 15

Mandrill Demo (DSP First)

• Precision– Input is represented as eight-bit numbers [0,255] per image

pixel (i.e. fewer than three decimal digits of accuracy)

– Filter coeffients represented by one decimal digit each

– Intermediate computations (filtering) in double-precision floating-point arithmetic (15-16 decimal digits of accuracy)

– Output is represented as eight-bit number [-128, 127](i.e. fewer than three decimal digits)

• No output precision was harmed in the making of this demo

5 - 16

Finite Impulse Response Filters

• Duration of impulse response h[k] is finite, i.e. zero-valued for k outside interval [0, M-1]:

– Output depends on current input and previous M-1 inputs– Summation to compute y[k] reduces to a vector dot product

between M input samples in the vector and M values of the impulse response in vector

• What instruction set architecture features would you add to accelerate FIR filtering?

1

0

M

mm

mkxmhmkxmhkhkxky

)]1([ ..., ],1[ ],[ Mkxkxkx

]1[ ..., ],1[ ],0[ Mhhh

5 - 17

Symmetric FIR Filters

• Impulse response often symmetric about midpoint– Phase of frequency response is linear (slides 5-9 to 5-11)

– Example: three-tap FIR filter (M = 3) with h[0] = h[2]

• Implementation savings– Reduce number of multiplications from M to M/2 for even-

length and to (M+1)/2 for odd-length impulse responses

– Reduce storage of impulse response by same amount

– TI TMS320C54 DSP has an accelerator instructor FIRS to compute h[0] ( x[k] + x[k-2] ) in one instruction cycle

– On most DSPs, no accelerator instruction is available

]1[ ]1[]2[][]0[

]2[ ]2[]1[ ]1[][ ]0[

kxhkxkxh

kxhkxhkxhky

5 - 18

Filter Design

• Specify a desired piecewise constant magnitude response

• Lowpass filter example [0, p], mag [1-p, 1]

[s, ], mag [0, s]Transition band unspecified

• Symmetric FIR filter design methodsWindowingLeast squaresRemez (Parks-McClellan)

1

s

p

p s

Desired Magnitude Response

Passband StopbandTransition band

Red region is forbidden

Lowpass Filter Example

p passband ripple s stopband ripple

forbidden

forbidden

forbidden

Achtung!

5 - 19

Importance of Linear phase

• Speech signalsUse phase differences to

locate a speaker

Once locked onto a speaker, our ears are relatively insensitive to phase distortion in speech from that speaker (underlies speech compression systems, e.g. digital cell phones)

• Linear phase crucialAudio

Images

Communication systems

• Linear phase response Need FIR filters

Realizable IIR filters cannot achieve linear phase response over all frequencies

5 - 20

Z-transform Definition

• For discrete-time systems, z-transforms play same role as Laplace transforms do for continuous-time systems

Inverse transform requires contour integration over closed contour (region) R

Contour integration covered in a Complex Analysis course

• Compute forward and inverse transforms using transform pairs and properties

k

kzkhzH )(

Bilateral Forward z-transform

R

k dzzzHj

kh 1 )( 2

1][

Bilateral Inverse z-transform

5 - 21

Common Z-transform Pairs

• h[k] = [k]

Region of convergence: entire z-plane

• h[k] = [k-1]

Region of convergence: entire z-plane

h[k-1] z-1 H(z)

1 0

0

k

k

k

k zkzkzH

11

1

1 1

zzkzkzHk

k

k

k

1 if 1

1

00

z

a

za

z

aza

zkuazH

k

k

k

kk

k

kk

• h[k] = ak u[k]

Region of convergence: |z| > |a|

|z| > |a| is the complement of a disk

5 - 22

Region of Convergence• Region of the complex z-

plane for which forward z-transform converges

Im{z}

Re{z}Entire plane

Im{z}

Re{z}Complement of a disk

Im{z}

Re{z}Disk

Im{z}

Re{z}

Intersection of a disk and complement of a disk

• Four possibilities (z=0 is a special case that may or may not be included)

5 - 23

azza

kuaZ

k

for 1

1

1

Stability

• Rule #1: For a causal sequence, poles are inside the unit circle (applies to z-transform functions that are ratios of two polynomials) OR

• Rule #2: Unit circle is in the region of convergence. (For continuous-time signal, imaginary axis would be in region of convergence of Laplace transform.)

• Example:

Stable if |a| < 1 by rule #1 or equivalently

Stable if |a| < 1 by rule #2 because |z|>|a| and |a|<1

5 - 24

Transfer Function

• Transfer function is z-transform of impulse response; e.g. for FIR filter with M taps (slide 5-3)

Region of convergence is entire z-plane

FIR filters are always stable

• Substitute z = e j into transfer function to obtain frequency responseValid when unit circle is in region of convergence (i.e. for

stable systems according to rule #2 on last slide)

)1(11

0

]1[ ]1[]0[ )(

MM

k

k

k

k zMhzhhzkhzkhzH ...

)1( 1

0

]1[ ]1[]0[ |)()( jMjM

k

kj

ez

j eMhehhekhzHeH j

...